Fragility Models

Introduction

Fragility functions describe the probability that a structure will reach or exceed a given damage state (DS) as a function of an intensity measure (IM). In this framework, fragility functions are assumed to follow a lognormal distribution, which is commonly adopted in seismic risk assessment due to its ability to capture uncertainty in structural response.

Damage State Definition

Four DSs are considered for structural fragility and the onset of these DSs is inferred from the maximum interstorey drift ratio (\(\theta_{max}\)) attaining pre-defined thresholds based on the capacity curve.

Description of the considered structural damage states and DS thresholds for MDOF systems expressed in terms of maximum interstorey drift ratio

Damage State	Label	Description	Maximum Interstorey Drift Ratio Threshold
DS1	Slight	Minor cracking, onset of yielding	\(\theta_{max,DS1} = \max_{i=1,...,n_s}[\Gamma_1(\phi_{1,i} - \phi_{1,i-1}) SD_y / h_i]\)
DS2	Moderate	Significant cracking, some spalling	\(\theta_{max,DS2} = \max_{i=1,...,n_s}[\Gamma_1(\phi_{1,i} - \phi_{1,i-1})(0.67 SD_y + 0.33 SD_u) / h_i]\)
DS3	Extensive	Major structural damage	\(\theta_{max,DS3} = \max_{i=1,...,n_s}[\Gamma_1(\phi_{1,i} - \phi_{1,i-1})(0.33 SD_y + 0.67 SD_u) / h_i]\)
DS4	Complete	Building at or near collapse	\(\theta_{max,DS4} = \max_{i=1,...,n_s}[\Gamma_1(\phi_{1,i} - \phi_{1,i-1}) SD_u / h_i]\)

where \(SD_y\) and \(SD_u\) are yield and ultimate spectral displacements, \(\Gamma_1\) is the first-mode participation factor, \(\phi_{1,i}\) is the first-mode shape ordinate, and \(h_i\) is the storey height.

Fragility Function Derivation

Lognormal Fragility Functions

Let \(P(DS \ge ds_i | IM)\) denote the probability that damage state \(ds_i\) is exceeded at a given intensity measure level IM. Under the lognormal assumption:

\[P(DS \ge ds_i | IM) = \Phi\left(\frac{\ln(IM) - \ln(\mu_{DS_i})}{\beta_{DS_i}}\right)\]

where:

• \(\mu_{DS_i}\) is the median intensity measure corresponding to damage state \(ds_i\)

• \(\beta_{DS_i}\) is the total logarithmic standard deviation (dispersion)

• \(\Phi(\cdot)\) denotes the standard normal cumulative distribution function

Total Dispersion

The total dispersion accounts for multiple sources of uncertainty:

\[\beta_{DS_i} = \sqrt{\beta_{r2r}^2 + \beta_{b2b}^2 + \beta_{ds}^2}\]

where:

• \(\beta_{r2r}\) (record-to-record or \(\beta_{EDP|IM}\)) represents variability due to ground motion record-to-record uncertainty

• \(\beta_{b2b}\) (building-to-building) captures variability in structural properties across nominally similar buildings

• \(\beta_{ds}\) represents uncertainty associated with damage state threshold definition

A building-to-building dispersion of 0.30 is adopted following FEMA P-58 recommendations. A DS uncertainty of 0.30 to 0.40 is adopted following FEMA P-58 recommendations.

Collapse Fragility

In addition to the four damage states, a separate collapse fragility function is derived using logistic regression on collapse/non-collapse outcomes from the modified cloud analysis:

\[P(C | IM) = \frac{1}{1 + \exp[-(\alpha_0 + \alpha_1 \ln(IM))]}\]

The collapse cases are identified when:

• Numerical non-convergence occurs during NLTHA

• The EDP exceeds a collapse threshold \(EDP_C = 1.5 \times \theta_{max,DS4}\)

Interactive Viewer

Use the interactive viewer below to explore fragility functions for different building classes.

Model: GEM

Building Class: -- Select Building Class --

Region: -- Select Region --

IMT: -- Select IMT -- PGA SA(0.3) SA(0.6) SA(1.0)

Occupancy: -- Select Occupancy -- Residential (RES) Commercial (COM) Industrial (IND)

Add to Plot Clear All

X-Axis Max (g): 5.0

Fragility Parameters Summary

The table below shows the fragility function parameters for all building classes using the most efficient intensity measure (IM). Parameters include median values (θ) for each damage state and the total dispersion (β).

Per-IMT fragility parameters (medians and dispersions) are available in: {summaries}/structural_fragility_params_{IMT}.csv

Select an intensity measure below to browse the corresponding table.

Intensity Measure: PGA SA(0.3s) SA(0.6s) SA(1.0s)

Loading fragility parameters table...

Column Descriptions

Fragility Parameters Table Columns

Column	Description
Building Class	GEM taxonomy string identifying the building class
Efficient IM	Most efficient intensity measure for this building class
θ_DS1 to θ_DS4	Median IM values [g] for damage states DS1 through DS4
β_DS1 to β_DS4	Total logarithmic standard deviation for each damage state

References

• Baker JW. Efficient Analytical Fragility Function Fitting Using Dynamic Structural Analysis. Earthquake Spectra. 2015;31(1):579-599. doi:10.1193/021113EQS025M

• Lallemant, D., Kiremidjian, A., and Burton, H. (2015), Statistical procedures for developing earthquake damage fragility curves. Earthquake Engng Struct. Dyn., 44, 1373–1389. doi: 10.1002/eqe.2522.

• Jalayer, F., De Risi, R. & Manfredi, G. Bayesian Cloud Analysis: efficient structural fragility assessment using linear regression.

Bull Earthquake Eng 13, 1183–1203 (2015). https://doi.org/10.1007/s10518-014-9692-z

• Jalayer F, Ebrahimian H, Miano A, Manfredi G, Sezen H. Analytical fragility assessment using unscaled ground motion records.

Earthquake Engng Struct Dyn. 2017;46:2639–2663. https://doi.org/10.1002/eqe.2922